On the variety of Lagrangian subalgebras, II

Abstract

Motivated by Drinfeld's theorem on Poisson homogeneous spaces, we study the variety L of Lagrangian subalgebras of g ⊕ g for a complex semi-simple Lie algebra g . Let G be the adjoint group of g . We show that the ( G × G ) -orbit closures in L are smooth spherical varieties. We also classify the irreducible components of L and show that they are smooth. Using some methods of M. Yakimov, we give a new description and proof of Karolinsky's classification of the diagonal G-orbits in L , which, as a special case, recovers the Belavin–Drinfeld classification of quasi-triangular r -matrices on g . Furthermore, L has a canonical Poisson structure, and we compute its rank at each point and describe its symplectic leaf decomposition in terms of intersections of orbits of two subgroups of G × G .